20

JOHN AITCHISON

In the limit, as

dt

---* 0,

the differential perturbation process

{14.2)

is equivalent to

the set of differential equations

{14.4)

dt

d

log

Xi((t))

=

8i(t)- 8;(t) (i j).

Xj

t

There is redundancy in this set of!

D(D -1)

differential equations; the reduced set

{14.5)

d Xi(t)

-d log-(-)

=

8i(t)- 8n(t),

i

=

1, ... ,d,

t xn t

is sufficient to describe the process. Given the initial composition

x(O)

at time

0

we can readily obtain the composition

x(t)

at timet for the process by solving the

differential equations

{14.6)

x(t)

=

x(O)

o

P(t),

where the

ith

component

~(t)

of the overall perturbation

P(t)

is

(14.7)

~(t)

= exp

{lot

8i(u)du}.

We note that

{14.6)

and

{14.7)

can be expressed in terms of log ratios:

{14.8)

log{xi(t)/xn(t)}

=

log{xi(O)/xn(O)}

+

8i(t)- 8n(t),

i

=

1, ... , d,

and so we may anticipate that log ratio analysis of compositional data sets will

have a central role to play in the investigation of differential perturbation processes.

For purposes of exposition we have restricted our use of differential perturbation

processes to the case of a single continuous process variable

t.

No such restric-

tions are necessary. There may be a number of subprocesses, each with its own

driving continuous variable, in which case the differential equations defining the

process will involve partial derivatives. For compositional data sets with no contin-

uous variable defined, a log contrast principal component analysis will determine

the dimensionality of any such underlying continuous vector. Investigation of the

constant log contrast principal components is then the first step in any attempt to

identify the underlying processes. For some simple applications to sedimentary and

olivine compositions, see

[A-T].

15. Discussion

The history of simplicial inference abounds with all sorts of devices purport-

ing to overcome the difficulties associated with the special nature of the simplex

sample space. These range from imagining that there is no problem, ignoring the

constraint and inappropriately applying standard unconstrained

RD

multivariate

analysis; 'opening out' the compositions into imaginary vectors in

R!;_;

recognizing

that corr(xi,x;) = 0 does not mean that components

Xi

and

x;

are independent

but attempting to discover a non-zero 'null correlation' which characterizes inde-

pendence in imagined open vectors; studying compositional pathology in the sense

of identifying what goes

~rong

in applying standard multivariate analysis to sim-

plicial data; transforming from the simplex to the positive orthant of the sphere

and imagining that distributions over the whole sphere are appropriate for further

analysis; attempting to rid the Dirichlet class of its inherent independence proper-

ties. See

[Al2]

for further discussion of these attempted solutions.